Questions (33218 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Pre-U Pre-U 9794/3 2018 June Q5
9 marks Standard +0.3
5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, \(5 \%\) of the bottles contain more than 500 ml and \(2.5 \%\) contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.
  1. Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.
  2. It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.
Pre-U Pre-U 9794/3 2018 June Q6
12 marks Moderate -0.3
6 A volleyball squad has 11 players. A volleyball team consists of 6 players.
  1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
  2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
  3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.
Pre-U Pre-U 9794/3 2018 June Q7
5 marks Moderate -0.8
7 A particle is projected with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle 0.4 s after projection.
Pre-U Pre-U 9794/3 2018 June Q8
7 marks Easy -1.2
8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find
  1. the greatest height above the ground reached by the ball,
  2. the time taken for the ball to reach the ground.
Pre-U Pre-U 9794/3 2018 June Q9
5 marks Moderate -0.8
9 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. \(P\) collides directly with a stationary particle \(Q\) of mass 0.5 kg . This collision reverses the direction of motion of \(P\). Immediately after the collision the speed of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the value of \(m\),
  2. the coefficient of restitution between the two particles.
Pre-U Pre-U 9794/3 2018 June Q10
7 marks Standard +0.3
10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).
  1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
  2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.
Pre-U Pre-U 9794/3 2018 June Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-4_285_700_1043_721} Three forces are acting on a particle \(A\) as shown in the diagram. The forces act in the same plane and the particle is in equilibrium.
  1. Evaluate \(P\) and \(\theta\). The 8 N force is removed.
  2. State the direction of the instantaneous acceleration of \(A\).
Pre-U Pre-U 9794/3 2018 June Q12
12 marks Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.
  1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
  2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.
Pre-U Pre-U 9794/1 2018 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{69214874-18a7-495d-892d-2a0a7019cbe9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).
  1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
  2. Calculate the exact area of the region bounded by the curve and the tangent.
Pre-U Pre-U 9795/1 2019 Specimen Q3
2 marks Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
  2. State the value of \(k\) for which these three planes do not meet at a single point.
  3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).
Pre-U Pre-U 9795/1 2019 Specimen Q4
3 marks Challenging +1.2
4
  1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
  2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).
Pre-U Pre-U 9795/1 2019 Specimen Q6
5 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
  2. Deduce the coordinates of the turning points on \(C\).
  3. Sketch \(C\).
Pre-U Pre-U 9795/1 2019 Specimen Q7
2 marks Challenging +1.2
7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).
  1. Determine \(\mathrm { f } ^ { \prime \prime } ( 1 )\).
  2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
  3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
  4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2019 Specimen Q8
5 marks Challenging +1.8
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .
Pre-U Pre-U 9795/1 2019 Specimen Q9
3 marks Challenging +1.2
9
  1. Show that the substitution \(u = \frac { 1 } { y ^ { 3 } }\) transforms the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\) into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x$$
  2. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\), given that \(y = \frac { 1 } { 2 }\) when \(x = 0\). Give your answer in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2019 Specimen Q10
8 marks Standard +0.3
10 The line \(L\) has equation \(\mathbf { r } = \left( \begin{array} { c } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 4 \\ 6 \end{array} \right)\) and the plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { c } 2 \\ - 6 \\ 3 \end{array} \right) = k\).
  1. Given that \(L\) lies in \(\Pi\), determine the value of \(k\).
  2. Find the coordinates of the point, \(Q\), in \(\Pi\) which is closest to \(P ( 10,2 , - 43 )\). Deduce the shortest distance from \(P\) to \(\Pi\).
  3. Find, in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers, an equation for the plane which contains both \(L\) and \(P\).
Pre-U Pre-U 9795/1 2019 Specimen Q11
8 marks Challenging +1.8
11
  1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } } .$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
  2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
    1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.[3]
Pre-U Pre-U 9795/1 2019 Specimen Q12
6 marks Challenging +1.8
12
  1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 } .$$
  2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
    1. Sketch this curve.
    2. Find the exact length of the curve.
Pre-U Pre-U 9795/2 2019 Specimen Q1
4 marks Standard +0.3
1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.
  1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
  2. Find the value of \(\mathrm { E } ( X )\).
Pre-U Pre-U 9795/2 2019 Specimen Q2
5 marks Standard +0.3
2
  1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
  2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).
Pre-U Pre-U 9795/2 2019 Specimen Q3
4 marks Standard +0.3
3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below. $$\begin{array} { l l l l l l } 1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77 \end{array}$$
  1. Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  2. If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  3. If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (a) and (b).
Pre-U Pre-U 9795/2 2019 Specimen Q4
3 marks Standard +0.3
4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.
  1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
  2. Show that \(\operatorname { Var } ( \overline { a X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
  3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).
Pre-U Pre-U 9795/2 2019 Specimen Q5
3 marks Standard +0.3
5 The random variable \(X\) has probability density function \(\mathrm { f } ( \mathrm { x } )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.
  1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
  2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Show that the moment generating function of \(- X\) is \(k ( k + t ) ^ { - 1 }\).
  4. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Use the moment generating function of \(X _ { 1 } - X _ { 2 }\) to find the value of \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).
Pre-U Pre-U 9795/2 2019 Specimen Q6
2 marks Standard +0.8
6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 , \\ 0 & \text { otherwise } . \end{array} \right.$$
  1. Draw a sketch of this probability density function.
  2. Calculate the mean and the mode of \(X\).
  3. Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
  4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.
Pre-U Pre-U 9795/2 2019 Specimen Q7
4 marks Moderate -0.5
7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).